Let $S_g$ denote an ortientable surface of genus $g$. Let $\operatorname{Diff}(S_g)$ denote the group of diffeomorphism (that need not fix the orientation). Is there a name for the image of $\operatorname{Diff}(S_g) \to \operatorname{Aut}(H_1(S_g))=GL_{2n}(\mathbb Z)$? It is called symplectic group if we restrict to diffeomorphisms that preserve the orientation.
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1$\begingroup$ Do you really need a special name? You can just say that your favourite letter is a group preserving symplectic form up to sign and use that letter as a name. Signed symplectic group is one of you really need one. $\endgroup$– Denis TCommented Oct 21, 2022 at 17:40
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7$\begingroup$ I think it is sometimes written $GSp_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes. $\endgroup$– Oscar Randal-WilliamsCommented Oct 21, 2022 at 21:11
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$\begingroup$ @OscarRandal-Williams, since that seems (by my understanding and by voting!) clearly to be the right answer, maybe you could post it as such so that it can be accepted? $\endgroup$– LSpiceCommented Oct 22, 2022 at 17:01
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I think it is sometimes written $\operatorname{GSp}_{2g}(\mathbb{Z})$, and called the group of symplectic similitudes
answered Oct 22, 2022 at 17:40